Proximal Mappings Involving Almost Structured Matrices

We consider a minimization problem where the cost function consists of the sum of a quadratic data fidelity term and a penalty term. The quadratic involves a matrix H that can be embedded into a larger matrix (H) over tilde where multiplication with the inverse of I + alpha(H) over tilde (T)(H) over tilde can be efficiently performed. We discuss how to take advantage of this property when the Douglas-Rachford algorithm is utilized.